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Properties

Label 1-215-215.27-r0-0-0
Degree $1$
Conductor $215$
Sign $0.996 + 0.0775i$
Analytic cond. $0.998455$
Root an. cond. $0.998455$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 + 0.900i)2-s + (0.433 − 0.900i)3-s + (−0.623 + 0.781i)4-s + 6-s − i·7-s + (−0.974 − 0.222i)8-s + (−0.623 − 0.781i)9-s + (0.623 + 0.781i)11-s + (0.433 + 0.900i)12-s + (0.974 + 0.222i)13-s + (0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + (0.974 − 0.222i)17-s + (0.433 − 0.900i)18-s + (0.623 − 0.781i)19-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (0.433 − 0.900i)3-s + (−0.623 + 0.781i)4-s + 6-s − i·7-s + (−0.974 − 0.222i)8-s + (−0.623 − 0.781i)9-s + (0.623 + 0.781i)11-s + (0.433 + 0.900i)12-s + (0.974 + 0.222i)13-s + (0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + (0.974 − 0.222i)17-s + (0.433 − 0.900i)18-s + (0.623 − 0.781i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(215\)    =    \(5 \cdot 43\)
Sign: $0.996 + 0.0775i$
Analytic conductor: \(0.998455\)
Root analytic conductor: \(0.998455\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{215} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 215,\ (0:\ ),\ 0.996 + 0.0775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.607958788 + 0.06243172331i\)
\(L(\frac12)\) \(\approx\) \(1.607958788 + 0.06243172331i\)
\(L(1)\) \(\approx\) \(1.406604062 + 0.1558791871i\)
\(L(1)\) \(\approx\) \(1.406604062 + 0.1558791871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.433 + 0.900i)T \)
3 \( 1 + (0.433 - 0.900i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (0.974 + 0.222i)T \)
17 \( 1 + (0.974 - 0.222i)T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (0.781 - 0.623i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.974 + 0.222i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 + (0.781 + 0.623i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.974 - 0.222i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.433 + 0.900i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.781 + 0.623i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.98498194614669176695812992421, −25.65222959343421987382204722823, −24.83493478574754456261604355514, −23.53784266011325446957784955222, −22.42744691796222880136607639161, −21.85171881258429740305023496089, −20.961831093651137740229515698308, −20.34752716122850551429734479269, −19.028466803391765193958480071026, −18.63509142872855577978499050528, −17.001450431765622673408414704289, −15.81990705409442008024954393318, −14.92276658458810662337138919192, −14.10809107197201568806103237682, −13.099305638676302782973847474641, −11.783013410673924335701095451082, −11.1074017600512266557980368514, −9.91506465446276069944881835810, −9.08662100126625028833602030132, −8.23228066384137297733390396428, −5.90626482050880790292073822692, −5.29326317979685184940969703766, −3.70733651317645840428199077151, −3.192144293573539338481328448145, −1.64977965896524570675458043114, 1.2244222333004568202175691792, 3.14989022995426530666831951983, 4.16964345406454877944814648198, 5.63546127198264008248704978002, 6.987976209203226650231770651133, 7.26152882666607346169586366605, 8.55847176166664731687112781197, 9.51334514402515800023778518789, 11.27595040803943655794398968541, 12.49393594296009258242820371998, 13.27290781729287465113663376552, 14.165073964765937874003767064960, 14.793142359207230032443384829460, 16.17110546603023023067439152166, 17.10137764470980297419638486419, 17.93688615170572087868304982928, 18.85997547214468231003408266405, 20.16335291952992014639377936756, 20.8397819783631413169745365887, 22.343480954918075320126553530854, 23.2726497903653473285261495633, 23.720402120944159486427541563443, 24.80853582196893649601678421757, 25.5648752068985333319273307984, 26.23749109864590926096393473336

Graph of the $Z$-function along the critical line